Towards an Intelligent Tutor for Mathematical Proofs

Computer-supported learning is an increasingly important form of study since it allows for independent learning and individualized instruction. In this paper, we discuss a novel approach to developing an intelligent tutoring system for teaching textbook-style mathematical proofs. We characterize the particularities of the domain and discuss common ITS design models. Our approach is motivated by phenomena found in a corpus of tutorial dialogs that were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor for textbook-style mathematical proofs can be built on top of an adapted assertion-level proof assistant by reusing representations and proof search strategies originally developed for automated and interactive theorem proving. The resulting prototype was successfully evaluated on a corpus of tutorial dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453

Calculational Proofs in ACL2s

Teaching college students how to write rigorous proofs is a critical objective in courses that introduce formal reasoning. Over the course of several years, we have developed a mechanically-checkable style of calculational reasoning that we used to teach over a thousand freshman-level undergraduate students how to reason about computation in our "Logic and Computation" class at Northeastern University. We were inspired by Dijkstra, who advocated the use of calculational proofs, writing "calculational proofs are almost always more effective than all informal alternatives, ..., the design of calculational proofs seems much more teachable than the elusive art of discovering an informal proof." Our calculational proof checker is integrated into ACL2s and is available as an Eclipse IDE plugin, via a Web interface, and as a stand-alone tool. It automatically checks proofs for correctness and provides useful feedback. We describe the architecture of the checker, its proof format, its underlying algorithms, its correctness and provide examples using proofs from our undergraduate class and from Dijkstra. We also describe our experiences using the proof checker to teach undergraduates how to formally reason about computation

On the readability of machine checkable formal proofs

It is possible to implement mathematical proofs in a machine-readable language. Indeed, certain proofs, especially those deriving properties of safety-critical systems, are often required to be checked by machine in order to avoid human errors. However, machine checkable proofs are very hard to follow by a human reader. Because of their unreadability, such proofs are hard to implement, and more difficult still to maintain and modify. In this thesis we study the possibility of implementing machine checkable proofs in a more readable format. We design a declarative proof language, SPL, which is based on the Mizar language. We also implement a proof checker for SPL which derives theorems in the HOL system from SPL proof scripts. The language and its proof checker are extensible, in the sense that the user can modify and extend the syntax of the language and the deductive power of the proof checker during the mechanisation of a theory. A deductive database of trivial knowledge is used by the proof checker to derive facts which are considered trivial by the developer of mechanised theories so that the proofs of such facts can be omitted. We also introduce the notion of structured straightforward justifications, in which simple facts, or conclusions, are justified by a number of premises together with a number of inferences which are used in deriving the conclusion from the given premises. A tableau prover for first-order logic with equality is implemented as a HOL derived rule and used during the proof checking of SPL scripts. The work presented in this thesis also includes a case study involving the mechanisation of a number of results in group theory in SPL, in which the deductive power of the SPL proof checker is extended throughout the development of the theory

Axmedis 2005

The AXMEDIS conference aims to promote discussions and interactions among researchers, practitioners, developers and users of tools, technology transfer experts, and project managers, to bring together a variety of participants. The conference focuses on the challenges in the cross-media domain (which include production, protection, management, representation, formats, aggregation, workflow, distribution, business and transaction models), and the integration of content management systems and distribution chains, with particular emphasis on cost reduction and effective solutions for complex cross-domain problems